# How many digits?

• Apr 18th 2011, 03:11 PM
sundance240
How many digits?
How many digits does the number 2^100 have? Using complicated caculator softwares it is 31 digits but how will you find out with pencil and paper
• Apr 18th 2011, 05:18 PM
Soroban
Hello, sundance240!

Quote:

How many digits does the number 2^100 have?
Using complicated caculator softwares, it has 31 digits,
but how will you find out with pencil and paper?

We note that: .2^{10} .= .1024 . .10^3

Hence: .2^{100} . .(10^3)^{10} .= .10^{30}

. . which has 31 digits.

• Apr 19th 2011, 12:09 AM
Opalg
Quote:

Originally Posted by sundance240
How many digits does the number 2^100 have? Using complicated caculator softwares it is 31 digits but how will you find out with pencil and paper

In prehistoric times we used to do problems like this by using logarithms to base 10. These were used so often that everybody knew that log(2) = 0.3010... . Therefore log(2^{100}) = 100*log(2) = 30.10..., and since this is between 30 and 31 it follows that 2^{100} is a 31-digit number.
• Apr 19th 2011, 01:06 AM
emakarov
Quote:

Originally Posted by Opalg
In prehistoric times we used to do problems like this by using logarithms to base 10. These were used so often that everybody knew that log(2) = 0.3010... .

Wow...
• Apr 19th 2011, 04:21 AM
topsquark
Quote:

Originally Posted by Opalg
In prehistoric times we used to do problems like this by using logarithms to base 10. These were used so often that everybody knew that log(2) = 0.3010... . Therefore log(2^{100}) = 100*log(2) = 30.10..., and since this is between 30 and 31 it follows that 2^{100} is a 31-digit number.

Wow! What a dinosaur. I suppose Napier's bones are fossils by now? (Rofl)

-Dan
• Apr 19th 2011, 05:34 AM
Deveno
Quote:

Originally Posted by Opalg
In prehistoric times we used to do problems like this by using logarithms to base 10. These were used so often that everybody knew that log(2) = 0.3010... . Therefore log(2^{100}) = 100*log(2) = 30.10..., and since this is between 30 and 31 it follows that 2^{100} is a 31-digit number.

of course, the "log" button on any reputable brand of calculator (including the generic one shipped with any version of Windows) will accomplish the same thing.
• Apr 19th 2011, 11:51 AM
HallsofIvy
Quote:

Originally Posted by topsquark
Wow! What a dinosaur. I suppose Napier's bones are fossils by now? (Rofl)

-Dan

Years "B.C."- before calculators.